Mathematical Modelling of Sediment Thickness and Bedrock Topography Using the Biharmonic Equation and Constrained Sequential Gaussian Simulation: Bi-cSGS-Surface-v1.0

Abstract. In this study, we present a two-stage surface reconstruction framework in which the target surface is modeled as a spatial random function composed by a trend function and stochastic residual component. The trend is recovered by solving the discrete form of a biharmonic equation, ensuring a smooth representation of the large scale structure, while the residual is simulated using constrained Sequential Gaussian Simulation (cSGS) to reproduce small scale spatial variability. Bounds are imposed during simulation to maintain consistency with geomorphological constraints.

The approach is implemented using measurement data from wells, along with Digital Elevation Models and digital quaternary maps as input. Two strategies are evaluated: cSGS applied directly to the observed variables, and cSGS applied to the residuals after subtracting the biharmonic trend. Variogram models are fitted to experimental variograms obtained from the data, and model performance is assessed using an independent dataset. Cross validation on an independent dataset shows that explicitly incorporating the trend component prior to stochastic simulation improves predictive accuracy, yielding lower mean absolute error and stochastic simulation results better centered around observations.

The methodology demonstrates the importance of modeling large-scale trends before cSGS and provides a flexible, physically consistent framework for estimating sediment thickness and bedrock topography under sparse data and uncertainty.